Soliton solutions of the Toda hierarchy on quasi-periodic backgrounds revisited (original) (raw)

Soliton defects in one-gap periodic system and exotic supersymmetry

Physical Review D, 2014

By applying Darboux-Crum transformations to the quantum one-gap Lamé system, we introduce an arbitrary countable number of bound states into forbidden bands. The perturbed potentials are reflectionless and contain two types of soliton defects in the periodic background. The bound states with finite number of nodes are supported in the lower forbidden band by the periodicity defects of the potential well type, while the pulse type bound states in the gap have infinite number of nodes and are trapped by defects of the compression modulations nature. We investigate the exotic nonlinear N = 4 supersymmetric structure in such paired Schrödinger systems, which extends an ordinary N = 2 supersymmetry and involves two bosonic generators composed from Lax-Novikov integrals of the subsystems. One of the bosonic integrals has a nature of a central charge, and allows us to liaise the obtained systems with the stationary equations of the Korteweg-de Vries and modified Korteweg-de Vries hierarchies. This exotic supersymmetry opens the way for the construction of self-consistent condensates based on the Bogoliubov-de Gennes equations and associated with them new solutions to the Gross-Neveu model. They correspond to the kink or kink-antikink defects of the crystalline background in dependence on whether the exotic supersymmetry is unbroken or spontaneously broken.

Propagation of solitons in the Toda lattice with an impure segment

Physical review, 2000

The transmission and scattering of a single soliton is studied numerically in the Toda lattice with an impure segment which consists of two kinds of masses. The incident soliton is split into transmitted, reflected, and trapped solitons by the impure segment. The energy of the soliton trapped in the segment escapes from the segment very slowly and thus we can define the transmission rate by the ratio of energies of the transmitted soliton and the incident soliton. It is shown that the dependence of the transmission rate on the segment length N can be fitted quite well by 1/(1ϩ␣N ␤). The transmission rate is also shown to be a monotone decreasing function of the wave number of the incident soliton. Most of the energy of the transmitted wave is carried by a large soliton ͑the frontier soliton͒ at the front, which is shown to be an exact soliton of the Toda lattice. When the mass difference is small, the transmission rate can be obtained by considering the segment as a repetition of a unit and repeating the renormalization of the wave number due to the unit.

Generalised Fourier transform and perturbations to soliton equations

Discrete & Continuous Dynamical Systems - B, 2009

A brief survey of the theory of soliton perturbations is presented. The focus is on the usefulness of the so-called Generalised Fourier Transform (GFT). This is a method that involves expansions over the complete basis of "squared solutions" of the spectral problem, associated to the soliton equation. The Inverse Scattering Transform for the corresponding hierarchy of soliton equations can be viewed as a GFT where the expansions of the solutions have generalised Fourier coefficients given by the scattering data. The GFT provides a natural setting for the analysis of small perturbations to an integrable equation: starting from a purely soliton solution one can 'modify' the soliton parameters such as to incorporate the changes caused by the perturbation. As illustrative examples the perturbed equations of the KdV hierarchy, in particular the Ostrovsky equation, followed by the perturbation theory for the Camassa-Holm hierarchy are presented.

Spectral approach for the soliton and periodic solutions of the nonlinear wave equation

Journal of the Optical Society of America B, 1993

A spectral method that obtains the soliton and periodic solutions to the nonlinear wave equation is presented. The results show that the nonlinear group velocity is a function of the frequency shift as well as of the soliton power. When the frequency shift is a function of time, a solution in terms of the Jacobian elliptic function is obtained. This solution is periodic in nature, and, to generate such an optical pulse train, one must simultaneously amplitude-and frequency-modulate the optical carrier. Finally, we extend the method to include the effect of self-steepening.

On the periodic Toda lattice with a self-consistent source

Communications in Nonlinear Science and Numerical Simulation

This work is devoted to the application of inverse spectral problem for integration of the periodic Toda lattice with self-consistent source. The effective method of solution of the inverse spectral problem for the discrete Hill’s equation is presented.

Scattering theory for Jacobi operators with a steplike quasi-periodic background

Inverse Problems, 2007

We develop direct and inverse scattering theory for Jacobi operators with steplike quasi-periodic finite-gap background in the same isospectral class. We derive the corresponding Gel'fand-Levitan-Marchenko equation and find minimal scattering data which determine the perturbed operator uniquely. In addition, we show how the transmission coefficients can be reconstructed from the eigenvalues and one of the reflection coefficients.

Scattering Theory for Jacobi Operators with Quasi-Periodic Background

Communications in Mathematical Physics, 2006

We develop direct and inverse scattering theory for Jacobi operators which are short range perturbations of quasi-periodic finite-gap operators. We show existence of transformation operators, investigate their properties, derive the corresponding Gel'fand-Levitan-Marchenko equation, and find minimal scattering data which determine the perturbed operator uniquely.